Here is a partial answer. It proves a fault-free rectangle can be assembled from rectangles of size mxn such that one dimension is not a multiple of the other.
The remaining cases can be converted to the 1xn case solved earlier by Bubbler.
Using Jaap's interpretation of the problem, I believe
Proof by example:
Informal proof (might have a flaw):
A lemma is needed at this point:
Back to the second claim:
As a demonstration of the construction, here are 2×5 tiles covering a 30×22 rectangle and 3×4 tiles covering a 36×27 rectangle, "seamlessly". ...